Asymmetric Dependence in Finance E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Wiley Finance Editions
• Sprache: Englisch

Seitenzahl: 599

Veröffentlichungsjahr: 2018

Beschreibung

Avoid downturn vulnerability by managing correlation dependency Asymmetric Dependence in Finance examines the risks and benefits of asset correlation, and provides effective strategies for more profitable portfolio management. Beginning with a thorough explanation of the extent and nature of asymmetric dependence in the financial markets, this book delves into the practical measures fund managers and investors can implement to boost fund performance. From managing asymmetric dependence using Copulas, to mitigating asymmetric dependence risk in real estate, credit and CTA markets, the discussion presents a coherent survey of the state-of-the-art tools available for measuring and managing this difficult but critical issue. Many funds suffered significant losses during recent downturns, despite having a seemingly well-diversified portfolio. Empirical evidence shows that the relation between assets is much richer than previously thought, and correlation between returns is dependent on the state of the market; this book explains this asymmetric dependence and provides authoritative guidance on mitigating the risks. * Examine an options-based approach to limiting your portfolio's downside risk * Manage asymmetric dependence in larger portfolios and alternate asset classes * Get up to speed on alternative portfolio performance management methods * Improve fund performance by applying appropriate models and quantitative techniques Correlations between assets increase markedly during market downturns, leading to diversification failure at the very moment it is needed most. The 2008 Global Financial Crisis and the 2006 hedge-fund crisis provide vivid examples, and many investors still bear the scars of heavy losses from their well-managed, well-diversified portfolios. Asymmetric Dependence in Finance shows you what went wrong, and how it can be corrected and managed before the next big threat using the latest methods and models from leading research in quantitative finance.

Leseprobe

Table of Contents

Cover

Title Page

About the Editors

Introduction

NOTES

CHAPTER 1: Disappointment Aversion, Asset Pricing and Measuring Asymmetric Dependence

1.1 INTRODUCTION

1.2 FROM SKIADAS PREFERENCES TO ASSET PRICES

1.3 CONSISTENTLY MEASURING ASYMMETRIC DEPENDENCE

1.4 SUMMARY

REFERENCES

FURTHER READING

NOTES

CHAPTER 2: The Size of the CTA Market and the Role of Asymmetric Dependence

2.1 INTRODUCTION

2.2 MARKET MODEL

2.3 COMPUTATION OF MOMENTS

2.4 EXAMPLE DISTRIBUTIONS

2.5 HETEROGENEITY AND CTA MARKET SIZE

2.6 EMPIRICAL EXAMPLES

2.7 CONCLUSIONS

REFERENCES

NOTES

CHAPTER 3: The Price of Asymmetric Dependence

3.1 INTRODUCTION

3.2 THE ASYMMETRIC DEPENDENCE RISK PREMIUM

3.3 CONCLUSION

REFERENCES

FURTHER READING

NOTES

CHAPTER 4: Misspecification in an Asymmetrically Dependent World: Implications for Volatility Forecasting

4.1 INTRODUCTION

4.2 LITERATURE SURVEY

4.3 MODEL SPECIFICATIONS

4.4 ESTIMATING ‘TRUE’ PARAMETER VALUES

4.5 EVALUATING FORECASTING PERFORMANCE

4.6 SIMULATION METHOD AND RESULTS

4.7 CONCLUSION

REFERENCES

NOTES

Appendix 4.A: Additional Details Regarding Underlying Data Sources Used by Global Financial Data and Bloomberg

Appendix 4.B: Proof of Theorem 4.1

Appendix 4.C: Proof of Corollaries 4.1 and 4.2

CHAPTER 5: Hedging Asymmetric Dependence

5.1 INTRODUCTION

5.2 ASYMMETRIC DEPENDENCE IN IMPLIED EQUITY CORRELATION: THE IMPLIED CORRELATION SKEW

5.3 THE EFFECT OF CORRELATION SKEW ON PORTFOLIO CHOICE

5.4 EQUITY CORRELATION PRODUCTS

5.5 MODELS FOR CORRELATION SKEW

REFERENCES

NOTES

CHAPTER 6: Orthant Probability‐Based Correlation

6.1 INTRODUCTION

6.2 ORTHANT PROBABILITIES AND ORTHANT CORRELATION

6.3 ORTHANT PROBABILITY TESTING

6.4 CHARACTERISTICS OF ORTHANT CORRELATIONS

6.5 IN THE PRESENCE OF SKEWNESS AND KURTOSIS

6.6 QUANTIFYING THE COMPLEMENTARITY OF ASSET CHARACTERISTICS

6.7 CONCLUSIONS

REFERENCES

Appendix 6.A: Proof of Application of Sheppard's Theorem to the Bivariate Elliptical

CHAPTER 7: Risk Measures Based on Multivariate Skew Normal and Skew

‐Mixture Models

7.1 INTRODUCTION

7.2 FINITE MIXTURE OF SKEW DISTRIBUTIONS

7.3 LINEAR TRANSFORMATION OF SKEW NORMAL AND SKEW

‐MIXTURES

7.4 RISK MEASURES

7.5 APPLICATION TO AN AUSTRALIAN PORTFOLIO

7.6 SUMMARY AND CONCLUSIONS

REFERENCES

CHAPTER 8: Estimating Asymmetric Dynamic Distributions in High Dimensions

8.1 INTRODUCTION

8.2 SEQUENTIAL PROCEDURE

8.3 THEORETICAL MOTIVATION

8.4 PARAMETERIZATIONS

8.5 EMPIRICAL APPLICATION

8.6 CONCLUDING REMARKS

REFERENCES

NOTES

Appendix 8.A

CHAPTER 9: Asymmetric Dependence, Persistence and Firm‐Level Stock Return Predictability

9.1 PREDICTIVE POWER OF ASYMMETRIC DEPENDENCE

9.2 PERSISTENCE OF ASYMMETRIC DEPENDENCE

9.3 SPILLOVER EFFECTS

9.4 CONCLUSION

REFERENCES

NOTES

CHAPTER 10: The Most Entropic Canonical Copula with an Application to ‘Style’ Investment

10.1 INTRODUCTION

10.2 MAXIMUM ENTROPY AND COPULAS

10.3 PROPOSED METHOD

10.4 SIMULATION

10.5 APPLICATION TO ASSET ALLOCATION

10.6 CONCLUSION

REFERENCES

NOTES

Appendix 10.A: Basic Results

Appendix 10.B: APPROXIMATION OF POTENTIAL FUNCTIONS

Appendix 10.C: PROOFS

Appendix 10.D: ESTIMATION OF THE DYNAMIC MECC MODEL

CHAPTER 11: Canonical Vine Copulas in the Context of Modern Portfolio Management: Are They Worth It?

11.1 INTRODUCTION

11.2 DATA

11.3 RESEARCH METHOD

11.4 RESULTS

11.5 CONCLUSION

REFERENCES

NOTES

Index

End User License Agreement

List of Tables

Chapter 2

TABLE 2.1 Estimated market size data: futures positions are the largest absolute position size (long or short) in the weekly CFTC Commitments of Traders report during 2010 (gold is 304,564 lots, equivalent to 30,456,400 troy ounces; silver is 66,066 lots or 330,330,000 troy ounces); gold market size estimate is from World Gold Council (WGC, 2010) and silver from Thomson Reuters (GFMS, 2011)

Chapter 3

TABLE 3.1 This table presents the correlation between each factor. We restrict our attention to stocks listed on the NYSE between January 1963 and December 2015. At each month,

, we estimate

,

,

, idiosyncratic risk (‘Idio’), coskewness (‘Cosk’), cokurtosis (‘Cokurt’) and

estimated using the next 12 months of daily excess return data, and natural logarithm of size (‘Size’), book‐to‐market ratio (‘BM’) and the average past 12‐monthly excess return (‘Past Ret’) computed as at time

. Returns (‘Ret’) are estimated as the average of the next 12‐monthly excess return. We proxy the market portfolio with the CRSP value‐weighted return of all NYSE, AMEX and NASDAQ stocks and the risk‐free rate with the 1‐month T‐bill rate. All factors are winsorized at the 1% and 99% level at each month

TABLE 3.2 For a given month, we first sort stocks into

deciles and then into

deciles within each characteristic decile in Panel A. In Panels B and C, we first sort stocks into size or coskewness deciles, respectively, and then into

deciles within each characteristic decile. Dependence ranges from low (group 1) to high (group 10), which implies that

consists of stocks with high downside risk and

consists of stocks with high upside potential. We record and report the equal‐weighted average 12‐monthly excess return for all stocks within each group, expressed as an effective annual rate of return. We restrict our attention to stocks listed on the NYSE between January 1963 and December 2015. We proxy the market portfolio with the CRSP value‐weighted return of all NYSE, AMEX and NASDAQ stocks and the risk‐free rate with the 1‐month T‐bill rate. We provide the spread (‘Diff’) for each row and column, given by the return associated with the high‐risk group, less the return associated with the low‐risk group. We also include the average return (‘Avg’) for each row and column

TABLE 3.3 We measure risk premia using the in‐sample (Ang

et al.

, 2006a) regressions where cross‐sectional regressions are computed every month rolling forward. At a given month,

, the average of the next 12 excess monthly returns is regressed against

,

,

, idiosyncratic risk (‘Idio’), coskewness (‘Cosk’), cokurtosis (‘Cokurt’) and

, estimated using the next 12 months of daily excess return data, and size (‘Log‐size’), book‐to‐market ratio (‘BM’) and the average past 12‐monthly excess return (‘Past Ret’), computed as at time

. We proxy the market portfolio with the CRSP value‐weighted return of all NYSE, AMEX and NASDAQ stocks and the risk‐free rate with the 1‐month T‐bill rate. All regressors are winsorized at the 1% and 99% level at each month. We restrict our attention to stocks listed on the NYSE between January 1963 and December 2015. Statistical significance is determined using Newey and West 1987 adjusted

t

‐statistics, given in parentheses, to control for overlapping data using the Newey and West 1994 automatic lag selection method to determine the lag length. The mean and standard deviation (in parentheses) for each variable are provided in the last column. All coefficients are reported as effective annual rates

TABLE 3.4 This table presents out‐of‐sample (Fama and MacBeth, 1973) regression results where the averages of the next 1 month, 3 months, 6 months and 12 months of monthly excess returns are regressed upon past risk factors. At a given month,

, the average of the next excess monthly returns is regressed against

,

,

, idiosyncratic risk (‘Idio’), coskewness (‘Cosk’), cokurtosis (‘Cokurt’),

and

, estimated using the past 12 months of daily excess return data. We also include the average past 12‐monthly excess return (‘Past Ret’). The relevant book‐to‐market ratio (‘BM’) at time

for a given stock is computed using the last available (most recent) book value entry. Size (‘Log‐size’) is computed at the same date that the book‐to‐market ratio is computed. We proxy the market portfolio with the CRSP value‐weighted return of all NYSE, AMEX and NASDAQ stocks and the risk‐free rate with the 1‐month T‐bill rate. We restrict our attention to stocks listed on the NYSE between January 1963 and December 2015. Statistical significance is determined using Newey and West 1987 adjusted

t

‐statistics, given in parentheses, to control for overlapping data using the Newey and West 1994 automatic lag selection method to determine the lag length. The mean and standard deviation (in parentheses) for each variable are also provided. All coefficients are reported as effective annual rates

TABLE 3.5 We measure risk premia using in‐sample (Ang

et al.

, 2006a) regressions where cross‐sectional regressions are computed every month rolling forward. We provide regression results using all available observations, as well as a series of regressions excluding the top quintile, top decile and top vigintile of volatile stocks, where volatility is measured as the standard deviation of the past 12 months of daily excess returns. We proxy the market portfolio with the CRSP value‐weighted return of all NYSE, AMEX and NASDAQ stocks and the risk‐free rate with the 1‐month T‐bill rate. All regressors are winsorized at the 1% and 99% level at each month. We restrict our attention to stocks listed on the NYSE between January 1963 and December 2015. With the exception of the non‐overlapping data regression, statistical significance is determined using Newey and West 1987 adjusted

t

‐statistics, given in parentheses, to control for overlapping data using the Newey and West 1994 automatic lag selection method to determine the lag length. All coefficients are reported as effective annual rates

TABLE 3.6 We measure risk premia using in‐sample (Ang

et al.

, 2006a) regressions where cross‐sectional regressions are computed every month rolling forward. Risk factors are estimated each month rolling forward and are calculated using 60, 24 and 6 months' worth of daily data, 5 and 3 years' worth of weekly data and 5 years' worth of fortnightly data. We also include value‐weighted regression results and non‐overlapping data regression results. We proxy the market portfolio with the CRSP value‐weighted return of all NYSE, AMEX and NASDAQ stocks and the risk‐free rate with the 1‐month T‐bill rate. All regressors are winsorized at the 1% and 99% level at each month. We restrict our attention to stocks listed on the NYSE between January 1963 and December 2015. With the exception of the non‐overlapping data regression, statistical significance is determined using Newey and West 1987 adjusted

t

‐statistics, given in parentheses, to control for overlapping data using the Newey and West 1994 automatic lag selection method to determine the lag length. All coefficients are reported as effective annual rates. Note that for the value‐weighted regressions, we calculate the value‐weighted mean and value‐weighted standard deviation of each risk factor at each month and report the time‐series average value‐weighted mean and the time‐series average value‐weighted standard deviation

Chapter 4

TABLE 4.1 S&P 500 returns – normal distribution specification, EGARCH(1,2)

TABLE 4.2 S&P 500 –

t

‐distribution specification, EGARCH(1,2)

TABLE 4.3 Kurtosis of squared equity returns

TABLE 4.4 US 10yr bond returns – normal distribution specification, EGARCH (1,2)

TABLE 4.5 US 10yr bond returns –

t

‐distribution specification, EGARCH (1,2)

TABLE 4.6(a) GARCH forecast error results:

β

= 0.98

TABLE 4.6(b) SV forecast error results:

β

= 0.98

TABLE 4.7(a) GARCH forecast error results:

β

= 0.90

TABLE 4.7(b) SV forecast error results:

β

= 0.90

TABLE 4.8(a) GARCH forecast error results:

β

= 0.80

TABLE 4.8(b) SV forecast error results:

β

= 0.80

TABLE 4.9(a) GARCH forecast error results:

β

= 0.75

TABLE 4.9(b) SV forecast error results:

β

= 0.75

Chapter 6

TABLE 6.1 Results of various benchmark cases of dependency between two distributions,

X

and

Y

, comparing the Pearson's product‐moment correlation coefficient to the four quadrant orthant correlations for the bivariate joint distribution. The ability of orthant correlations to fully capture non‐linearities, in comparison with Pearson's product‐moment linear correlation, is especially apparent in the classic textbook case of

Y

=

X

2

where there is full dependence between

X

and

Y

but no linear correlation exists

TABLE 6.2 Summary of assessment of quadrant orthant correlation desirable changes for the particular case of a two‐asset portfolio for which holding the first asset is a fixed assumption and the second asset is a candidate for portfolio addition. The rationale for ‘desirability’ of change is likely dependent on the particular application in mind and the rationale for this particular case is described in the text

Chapter 7

TABLE 7.1 Summary statistics of the monthly returns of three Australian stocks for period of early

to mid‐

TABLE 7.2 Performance of various mixture models on estimating the

VaR of three Australian stocks. The backtesting and independence values refer to the

‐value of the respective tests. The empirical VaR is

TABLE 7.3 Performance of various mixture models on estimating the

VaR of three Australian stocks. The backtesting and independence values refer to the

‐value of the respective tests. The empirical VaR is

Chapter 8

TABLE 8.1 Summary statistics of the returns

TABLE 8.2 Maximum likelihood parameter estimates for marginal distributions (robust standard errors are in parentheses)

TABLE 8.3

‐values of F‐tests for serial correlation

TABLE 8.4 Maximum likelihood parameter estimates for pairwise copulas (robust standard errors are in parentheses)

TABLE 8.5 Maximum likelihood parameter estimates of the compounding functions for groups of three assets (standard errors omitted)

TABLE 8.6 Maximum likelihood parameter estimates of the compounding functions for groups of four assets (standard errors omitted)

TABLE 8.7 Maximum likelihood parameter estimates of the compounding functions for groups of five assets (standard errors omitted)

TABLE 8.8 Maximum likelihood parameter estimates of time‐varying five‐dimensional

‐copula for the returns (robust standard errors are in parentheses)

TABLE 8.9 Growth of the number of parameters in a single optimization problem for the conventional and for the sequential methods based on the

‐copula

Chapter 9

TABLE 9.1 We measure risk premia using Fama and MacBeth (1973) regressions estimated every month rolling forward. We use the next 1‐month monthly excess return as dependent variable. All regressors are winsorized at the 1% and 99% level at each month. We restrict our attention to stocks listed on the NYSE between January 1959 and December 2015, US REITs listed on the NYSE between January 1972 and December 2013, stocks listed on the ASX between June 1992 and June 2014, and UK stocks listed between January 1987 and May 2015, respectively. Statistical significance is determined using Newey and West (1987) adjusted

‐statistics, given in parentheses, to control for overlapping data using the Newey and West (1994) automatic lag selection method to determine the lag length. All coefficients are reported as effective annual rates

TABLE 9.2 We measure risk premia using the Fama and MacBeth (1973) asset‐pricing procedure where value‐weighted cross‐sectional regressions are computed every month rolling forward. At a given month,

, the average of the mean of the next 1, 3, 6, 9, 12 and 15 months of excess monthly returns is regressed against

, idiosyncratic risk (‘Idio’), coskewness (‘Cosk’), cokurtosis (‘Cokurt’),

and

estimated using the past 12 months of daily excess return data. We also include the average past 12‐monthly excess return (‘Past Ret’). The relevant book‐to‐market ratio (‘BM’) at time

for a given stock is computed using the last available (most recent) book value entry. Size (‘Log‐size’) is computed at the same date that the book‐to‐market ratio is computed. We proxy the market portfolio with the CRSP value‐weighted return of all NYSE, AMEX and NASDAQ stocks and the risk‐free rate with the 1‐month T‐bill rate. We restrict our attention to stocks listed on the NYSE between January 1959 and December 2015. Statistical significance is determined using Newey and West (1987) adjusted

‐statistics, given in parentheses, to control for overlapping data using the Newey and West (1994) automatic lag selection method to determine the lag length. All coefficients are reported as effective annual rates

TABLE 9.3 We measure risk premia using the Fama and MacBeth (1973) asset‐pricing procedure and 3‐month and 6‐month predictive regressions estimated every month rolling forward. At a given month,

, the average 1‐month, 3‐month and 6‐month excess monthly return is regressed against

,

,

, idiosyncratic risk (‘Idio’), coskewness (‘Cosk’), cokurtosis (‘Cokurt’) and

estimated using the previous 12 months of daily excess return data, size (‘Log‐size’), book‐to‐market ratio (‘BM’) and the average past 12‐monthly excess return (‘Past Ret’), computed as at time

. We proxy the market portfolio with the CRSP value‐weighted return of all NYSE, AMEX and NASDAQ stocks and the risk‐free rate with the 1‐month T‐bill rate. All regressors are winsorized at the 1% and 99% level at each month. We restrict our attention to REIT stocks listed on the NYSE between January 1972 and December 2013. Statistical significance is determined using Newey and West (1987) adjusted

‐statistics, given in parentheses, to control for overlapping data using the Newey and West (1994) automatic lag selection method to determine the lag length. All coefficients are reported as effective annual rates

TABLE 9.4 We measure risk premia using the Fama and MacBeth (1973) asset‐pricing procedure and 3‐month and 6‐month predictive regressions estimated every month rolling forward. At a given month,

, the average 1‐month, 3‐month and 6‐month excess monthly return is regressed against

,

,

, idiosyncratic risk (‘Idio’), coskewness (‘Cosk’), cokurtosis (‘Cokurt’),

and

estimated using the previous 12 months of daily excess return data, size (‘Log‐size’), book‐to‐market ratio (‘BM’) and the average past 12‐monthly excess return (‘Past Ret’), computed as at time

. We proxy the market portfolio with the CRSP value‐weighted return of all NYSE, AMEX and NASDAQ stocks and the risk‐free rate with the 1‐month T‐bill rate. All regressors are winsorized at the 1% and 99% level at each month. We restrict our attention to REIT stocks listed on the NYSE between January 1972 and December 2013. Statistical significance is determined using Newey and West (1987) adjusted

‐statistics, given in parentheses, to control for overlapping data using the Newey and West (1994) automatic lag selection method to determine the lag length. All coefficients are reported as effective annual rates

TABLE 9.5 We measure risk premia using the Fama and MacBeth (1973) asset‐pricing procedure where value‐weighted cross‐sectional regressions are computed every month rolling forward. At a given month,

, the average of the mean of the next 1, 3, 6, 9, 12 and 15 months of excess monthly returns is regressed against

, idiosyncratic risk (‘Idio’), coskewness (‘Cosk’), cokurtosis (‘Cokurt’),

and

estimated using the past 12 months of daily excess return data. We also include the average past 12‐monthly excess return (‘Past Ret’). The relevant book‐to‐market ratio (‘BM’) at time

for a given stock is computed using the last available (most recent) book value entry. Size (‘Log‐size’) is computed at the same date that the book‐to‐market ratio is computed. We proxy the market portfolio with the S&P ASX 200 index and the risk‐free rate with the 90‐day bank accepted bill rate. We restrict our attention to stocks listed on the ASX between June 1992 and June 2014. Statistical significance is determined using Newey and West (1987) adjusted

‐statistics, given in parentheses, to control for overlapping data using the Newey and West (1994) automatic lag selection method to determine the lag length. All coefficients are reported as effective annual rates

TABLE 9.6 We measure risk premia using the Fama and MacBeth (1973) asset‐pricing procedure where value‐weighted cross‐sectional regressions are computed every month rolling forward. At a given month,

, the average of the mean of the next 1, 3, 6, 9, 12 and 15 months of excess monthly returns is regressed against

, idiosyncratic risk (‘Idio’), coskewness (‘Cosk’), cokurtosis (‘Cokurt’),

and

estimated using the past 12 months of daily excess return data. We also include the average past 12‐monthly excess return (‘Past Ret’). The relevant book‐to‐market ratio (‘BM’) at time

for a given stock is computed using the last available (most recent) book value entry. Size (‘Log‐size’) is computed at the same date that the book‐to‐market ratio is computed. We proxy the market portfolio with the FTSE 100 index and the risk‐free rate with the 3‐month UK Treasury bill rate. We restrict our attention to UK stocks listed between January 1987 and May 2015. Statistical significance is determined using Newey and West (1987) adjusted

‐statistics, given in parentheses, to control for overlapping data using the Newey and West (1994) automatic lag selection method to determine the lag length. All coefficients are reported as effective annual rates

TABLE 9.7 The

statistic is estimated using a 12‐month rolling window of daily returns. We estimate the AR(10) model for each firm in the sample and calculate the proportion of firms with significant AR coefficients. We restrict our attention to stocks listed on the NYSE between January 1959 and December 2015 (Panel A), US REITs listed on the NYSE between January 1972 and December 2013 (Panel B), stocks listed on the ASX between June 1992 and June 2014 (Panel C), and UK stocks listed between January 1987 and May 2015 (Panel D), respectively

TABLE 9.8

is estimated using a 12‐month rolling window of daily returns. We calculate the probability of migrations between

and

using non‐overlapping data. We restrict our attention to stocks listed on the NYSE between January 1959 and December 2015, US REITs listed on the NYSE between January 1972 and December 2013, stocks listed on the ASX between June 1992 and June 2014, and UK stocks listed between January 1987 and May 2015, respectively

TABLE 9.9 We calculate the aggregate level of AD as the average value of

across all firms from each financial market. At a given month,

,

is estimated using the past 12 months of daily excess return data. The aggregate levels of AD are winsorized at the 1% and 99% level. We restrict our attention to dates when data from NYSE, ASX and UK stocks is available, which starts from June 1992 and ends in December 2014. This table presents coefficients from the VAR(2) model of Equation (9.5) estimated using maximum likelihood

TABLE 9.10 We calculate the aggregate level of AD as the average value of

across all firms from each financial market. At a given month,

,

is estimated using the past 12 months of daily excess return data. The aggregate levels of AD are winsorized at the 1% and 99% level. We restrict our attention to dates when data from NYSE, ASX and UK stocks is available, which starts from June 1992 and ends in December 2014. This table presents coefficients from the VAR(2) model of Equation (9.5) estimated using ML. We report the estimated coefficients (Est), standard error of estimates (SE) and

‐statistic (

‐stat)

Chapter 10

TABLE 10.1 Descriptive sample statistics

TABLE 10.2(a) Results obtained from the

fluctuation

tests for serial correlation of the transformed residuals:

TABLE 10.2(b) Results obtained from the

fluctuation

goodness‐of‐fit tests for the uniformity of the transformed residuals:

TABLE 10.3 Results of the out‐of‐sample copula fluctuation test

TABLE 10.4 Results of copula specification search using information criteria

TABLE 10.5 Descriptive statistics of realized portfolio returns

TABLE 10.6 Descriptive statistics of optimal portfolios (normal copula)

TABLE 10.7 Descriptive statistics of optimal portfolios (MECC)

TABLE 10.8 Explaining optimal portfolio weights

TABLE 10.9 Pairwise comparisons of the models' performance

TABLE 10.10 Results obtained from White's reality check

Chapter 11

TABLE 11.1 Input data descriptive statistics

TABLE 11.2 Unconditional sample correlations

TABLE 11.3 Out‐of‐sample copula‐based portfolio strategy descriptive statistics

TABLE 11.4 Out‐of‐sample risk‐adjusted performance of copula‐based portfolio strategies

TABLE 11.5 Portfolio re‐balancing analysis across out‐of‐sample copula‐based portfolio strategies

TABLE 11.6 Economic measures of out‐of‐sample performance of copula‐based portfolio strategies

TABLE 11.7 Value‐at‐Risk (VaR) backtests across copula‐based portfolio strategies

TABLE 11.8 Average annual out‐of‐sample performance differential between CVC‐S vs. MVN models

List of Illustrations

Chapter 1

FIGURE 1.1 Simulated densities of GKP utility functions calculated when returns are symmetrically distributed (MVN) and asymmetrically distributed. Non‐disappointment‐averse utility is described by the GKP utility function (1.1) with

and

. Skiadas disappointment‐averse utility is described with

and

. Each of these two utility functions are calculated for both AD and symmetric distributions for two different conditioning events,

and

. The event

is the event that the market return is less than the certainty‐equivalent market return,

, and event

is the event that the market return is lower than the certainty‐equivalent market return,

, less two market return standard deviations.

FIGURE 1.2 Scatter plot of simulated bivariate data with asymmetric dependence (a) and symmetric dependence (b) that is used to test different downside‐risk metrics. The

sample is a random draw of bivariate data

where

and

, where

, with

,

and

. In (a),

so the sample displays LTAD. In (b),

so no AD is present and

are bivariate normal with linear dependence equal to

. Higher LTAD is proxied by increasing

, and higher UTAD is proxied by decreasing

.

FIGURE 1.3 Estimates of linear dependence and AD. We estimate the CAPM beta, downside beta and the Clayton copula parameter using

simulated pairs of data

, where

, with

and

. Higher levels of linear dependence are incorporated with higher values of

and higher levels of LTAD are incorporated with higher levels of

. Figure parts (a), (d) and (g) provide estimates for varying levels of linear dependence but with no AD (

). Figure parts (b), (e) and (h) provide estimates for varying degrees of AD with constant linear dependence (

). Figure parts (c), (f) and (i) provide estimates for varying degrees of linear dependence with constant AD (

).

FIGURE 1.4

data transformations. To calculate the

statistic with a random sample,

, as in (a), we let

where

is the continuously compounded return on the

i

th asset,

is the continuously compounded return on the market and

. This transformation forces

, as in (b). We standardize the transformed data, yielding

and

in (c). Finally, we re‐transform the data to have

by letting

and

in (d). The solid line through the middle of each plot is given to illustrate how the linear trend changes with each transformation.

FIGURE 1.5 Estimates of linear dependence and AD. We estimate the

statistic (Hong

et al.

, 2007) and the adjusted

statistic using

simulated pairs of data

, where

, with

and

. Higher levels of linear dependence are incorporated with higher values of

and higher levels of LTAD are incorporated with higher levels of

. Figure parts (a) and (d) provide estimates for varying levels of linear dependence but with no AD (

). Figure parts (b) and (e) provide estimates for varying degrees of AD with constant linear dependence (

). Figure parts (c) and (f) provide estimates for varying degrees of linear dependence with constant AD (

).

Chapter 2

FIGURE 2.1 Correlation and beta for uniform distribution.

FIGURE 2.2 Simulated correlation and beta between stock and option when future price has a normal distribution.

FIGURE 2.3 Correlation and beta for scale gamma distribution.

FIGURE 2.4 Correlation and beta for Pareto distribution.

Chapter 3

FIGURE 3.1 Linear vs. asymmetric dependence. Scatter plot of simulated bivariate data with asymmetric dependence (a) and symmetric dependence (b). The dependence between

and

may be described by a linear component and a higher‐order component, reflecting differences in dependence across the joint return distribution. A joint distribution that displays larger dependence in one tail compared with the opposite tail is said to display asymmetric dependence. In the case of (a), the dependence in the lower tail is higher than that of the upper tail, which is characteristic of LTAD in particular.

FIGURE 3.2 Our monthly sample size. We restrict our attention to stocks listed on the NYSE between January 1963 and December 2015.

FIGURE 3.3 Actual and hypothetical distribution of the

. We focus on stocks listed on the NYSE between January 1963 and December 2015. At a given month,

, we estimate

using the next 12 months of daily excess return data. We proxy the market portfolio with the CRSP value‐weighted return of all NYSE, AMEX and NASDAQ stocks and the risk‐free rate with the 1‐month T‐bill rate. The histogram of all

observations is presented in (a). We include the distribution of the

, computed using simulated multivariate normal data, parameterized at each month in (b). The size of each sample is chosen to match the number of days in each 12‐month period. The vertical lines represent

cutoffs following a

distribution. A positive (negative)

is indicative of excess upside (downside) risk over and above the tail risk implied by ordinary

.

FIGURE 3.4 This figure depicts the median factor loading for

,

and

at a given month,

, between January 1989 and December 2015 using the next 12 months of daily excess returns. We proxy the market portfolio with the CRSP value‐weighted return of all NYSE, AMEX and NASDAQ stocks and the risk‐free rate with the 1‐month T‐bill rate. The estimate is calculated using all historical data up to, and including, time

.

FIGURE 3.5 This figure depicts the factor sensitivity using the Fama and MacBeth 1973 asset‐pricing procedure where cross‐sectional regressions are computed every month rolling forward. At a given month,

, the average of the next 12 excess monthly returns is regressed against

, idiosyncratic risk, coskewness, cokurtosis,

and

estimated using the next 12 months of daily excess return data, and size (‘Log‐size’), book‐to‐market ratio (‘BM’) and the average past 12‐monthly excess return (‘Past Ret’), computed as at time

. We proxy the market portfolio with the CRSP value‐weighted return of all NYSE, AMEX and NASDAQ stocks and the risk‐free rate with the 1‐month T‐bill rate. All regressors are winsorized at the 1% and 99% level at each month. We restrict our attention to stocks listed on the NYSE between January 1963 and December 2015. The ‘Premium’ for

and for

and the ‘Discount’ for

between January 1989 and December 2015 is given by the time‐series median factor sensitivity using all historical sensitivity estimates up to, and including, time

.

Chapter 4

FIGURE 4.1 Simulation schematic

Chapter 5

FIGURE 5.1 Subplots (a) and (b) depict the DAX 30 and Eurostoxx prices and daily log returns between 3 January 2015 and 30 January 2016. Subplots (c) to (f) depict the implied correlation during calm market conditions (March 2015) and during stressed market conditions (January 2016) for the DAX 30 and the Eurostoxx 50 computed using Equation (5.1) for a range of strikes and maturities.

FIGURE 5.2 Traditional mean‐variance frontier.

FIGURE 5.3 Mean‐variance frontier with global kurtosis.

FIGURE 5.4 Mean‐variance frontier with global kurtosis and AD.

Chapter 6

FIGURE 6.1 The relation between orthant correlations and orthant probabilities illustrated for the bivariate case of the return distributions of two financial assets, shown as a function of quadrant. A base‐case neutral condition might be considered that of a bivariate standard normal (mean zero, variance one) distribution for which occupancy in each quadrant would be 25%, translating to a probability of 0.25 and leading to an orthant correlation of zero for all quadrants. Occupancies which differ from this 25% occupancy base‐case of occupancy or probability will result in either increases or decreases in correlation, depending on the quadrant involved.

FIGURE 6.2 Orthant correlations of the four quadrants of joint bivariate distributions for various levels of non‐linearity. (a) Orthant correlations are shown as a function of skewness between a standard normal

N

(0,1) distribution (with zero skewness and zero excess kurtosis) and a distribution with mean zero, variance of one, excess kurtosis of 10 and skewness varying from −2.5 to 2.5. In all cases, linear correlation as measured by Pearson's product‐moment correlation coefficient is pre‐specified and fixed for each case of skewness at 0.5. Quadrant two and three correlations,

ρ

01

and

ρ

00

, appear linearly related to skewness and quadrant one and four correlations,

ρ

11

and

ρ

10

, inversely linearly related. (b) Orthant correlations are shown between a standard normal

N

(0,1) distribution (with zero skewness and zero excess kurtosis) and a distribution with mean zero, variance of one, skewness of zero and various levels of excess kurtosis between 0 and 20. In all cases, linear correlation as measured by Pearson's product‐moment correlation coefficient is fixed to 0.5.

FIGURE 6.3

Δ

ρ

Q

as defined by Equation (6.9), as a function of excess kurtosis and skewness for the case of a bivariate joint distribution of asset returns in which the first asset is a standard normal distribution with mean zero, unit variance, zero skewness and zero kurtosis. In all cases, linear correlation as measured by the Pearson's product‐moment correlation coefficient was fixed to 0.5. (a) The second distribution has zero mean, unit variance and excess kurtosis fixed at 10, while the skewness is varied. (b) The second distribution has zero mean, unit variance and skewness fixed at zero, while the excess kurtosis is varied.

FIGURE 6.4

Δ

ρ

Q

as defined by Equation (6.9), as a function of the mean value of portfolio returns for an equally weighted combination of assets. Each of the 1,000 individual data points consists of a non‐overlapping subset of 100 joint samples from a bivariate normal

N

(0,1) distribution (with zero skewness and zero excess kurtosis) and linear correlation (as measured by Pearson's product‐moment correlation coefficient) was fixed to 0.5 for the entire sample of 100,000 random joint samples. According to a least‐squares best fit to a line (shown in the figure) and also as a result of

t

‐tests, there are reliable indications of a significant relation between

Δ

ρ

Q

and portfolio returns.

FIGURE 6.5 Linear correlation as measured by Pearson's product‐moment correlation coefficient, as a function of the mean value of portfolio returns for an equally weighted combination of assets. Each of the 1,000 individual data points consists of a non‐overlapping subset of 100 joint samples from a bivariate normal

N

(0,1) distribution (with zero skewness and zero excess kurtosis) and linear correlation (as measured by Pearson's product‐moment correlation coefficient) was fixed to 0.5 for the entire sample of 100,000 random joint samples. Both by eye and according to a least‐squares best fit to a line (shown in the figure), there is no indication of a relation between Pearson's

r

and portfolio returns.

Chapter 7

FIGURE 7.1 Contours of the density of some bivariate CFUST distribution with

,

and other parameters as follows. For cases (a) to (f),

is the identity matrix. For cases (g) and (h),

has diagonal elements being

and off‐diagonal elements being

. The elements of

from left to right, top to bottom are given by (

,0,3.5,3), (3,0,0,4), (2,

,5,4), (2,

,5,

), (

,

,

,

), (2,

,5,0), (0,

,

,

) and (2.5,1,

,

), respectively, for cases (a) to (h).

FIGURE 7.2 Time series of a portfolio of three Australian stocks for the period of January 2000 to June 2013. The top three panels correspond to the return of each of three individual stocks: Flight Centre Limited (FLT; dashed line), Westpac Banking Corporation (WBC; dotted line), Australia and New Zealand Banking Group Limited (ANZ; dash-dot line). The bottom panel shows the aggregate return (solid line) overlaid by the three stocks.

FIGURE 7.3 Contours of the fitted models to the Australian stock data. Each column shows the bivariate plot of a pair of stocks in the portfolio. The top panel shows the contours given by the fitted CFUST model. The middle panel shows the contours given by the fitted CFUSN model. The bottom panel shows the contours given by the fitted FM‐MN model.

FIGURE 7.4 Histogram of the three stocks in the portfolio overlaid with the marginal density of the fitted models. The dashed, dotted, and dash‐dot curves are given by the fitted CFUST, CFUSN and FM‐MN models, respectively.

FIGURE 7.5 Histogram of the aggregate return of the portfolio. The overlaid lines are the density of

calculated based on the fitted models, where the dashed, dotted, and dash‐dot curve corresponds to the CFUST, CFUSN and FM‐MN model, respectively.

FIGURE 7.6 Estimates of VaR provided by various fitted models. The solid line corresponds to the empirical value, whereas the dashed, dotted, and dash‐dot line represent the estimate given by the fitted CFUST, CFUSN and FM‐MN model, respectively.

FIGURE 7.7 Estimates of TCE given by various skew mixture models. The solid line corresponds to the empirical value, whereas the dashed, dotted, and dash‐dot line represent the estimate given by the CFUST, CFUSN and FM‐MN models, respectively.

Chapter 8

FIGURE 8.1 Relative prices and returns dynamics for GE, MCD, MSFT, KO and PG from 3 January to 31 December 2007.

FIGURE 8.2 Histograms of the returns.

FIGURE 8.3 Pairwise scatter plots of marginal distributions and sample correlations.

FIGURE 8.4 Kolmogorov–Smirnov tests of marginal distributions (

‐values in parentheses).

FIGURE 8.5 Pairwise Kolmogorov–Smirnov tests of bivariate copula specification: first five pairs of GE, MCD, MSFT, KO and PG (

‐values in parentheses).

FIGURE 8.6 Pairwise Kolmogorov–Smirnov tests of bivariate copula specification: second five pairs of GE, MCD, MSFT, KO and PG (

‐values in parentheses).

Chapter 9

FIGURE 9.1 The autocorrelation function is computed using the

statistic computed on 12‐month non‐overlapping periods using daily excess returns. One lag represents a 12‐month period. At a given month,

,

is estimated using the previous 12 months of daily excess return data. We restrict our attention to stocks listed on the NYSE between January 1959 and December 2015 in panel (a), US REITs listed on the NYSE between January 1972 and December 2013 in panel (b), stocks listed on the ASX between June 1992 and June 2014 in panel (c), and UK stocks listed between January 1987 and May 2015 in panel (d), respectively.

FIGURE 9.2 We calculate the aggregate level of AD as the average value of

across all firms from each financial market. At a given month,

,

is estimated using the past 12 months of daily excess return data. The aggregate levels of AD are winsorized at the 1% and 99% level. We restrict our attention to dates when data from NYSE, ASX and UK stocks is available, which starts from June 1992 and ends in December 2014. This figure presents a time series of the development of the aggregate level of

for the four different financial markets.

Chapter 10

FIGURE 10.1 Approximate marginal densities (

C

(

u

, 1) and

C

(1,

v

) on the vertical axis) of the MECC, which are constructed from rank correlations,

S

= 0.4248,

1

= 0.4967,

2

= 0.5430 and

= −0.1520, are plotted against

u

and

v

(on the horizontal axis).

FIGURE 10.2 Approximate marginal densities (

C

(

u

, 1) and

C

(1,

v

) on the vertical axis) of the MECC, which are constructed from rank correlations,

S

= 0.4248,

1

= 0.4967,

2

= 0.5430 and

= −0.1520, are plotted against

u

and

v

(on the horizontal axis).

FIGURE 10.3 Asymmetric correlation between the excess returns on Russell 1000 Growth and Value. The horizontal axis shows cutoff quantiles, the vertical axis shows

exceedance

correlations between these returns, and

ρ

+

and

ρ

−

denote the positive and negative

exceedance

correlations, respectively.

FIGURE 10.4 Moving‐window asymmetric correlation between the excess returns on Russell 1000 Growth and Value. The horizontal axis shows time periods, the vertical axis shows

exceedance

correlations between these returns, and

ρ

+

and

ρ

−

denote the positive and negative

exceedance

correlations, respectively.

FIGURE 10.5 Moving‐window asymmetric correlation between the risk‐adjusted (or standardized) excess returns on Russell 1000 Growth and Value. The horizontal axis shows time periods, the vertical axis shows

exceedance

correlations between these returns, and

ρ

+

and

ρ

−

denote the positive and negative

exceedance

correlations, respectively.

FIGURE 10.6 Moving‐window rank correlations in the out‐of‐sample period (October 1999 to July 2006).

ρ

is the Spearman's rho,

ν

1

,

ν

2

,

κ

and

ϕ

are, as defined in Section 10.1, the second, third, fourth and fifth rank correlations, respectively. The horizontal axis shows the out‐of‐sample period, and the vertical axis shows rank correlations.

FIGURE 10.7 Optimal

unconstrained

normal and MECC portfolios weights for an investor with degree of relative risk aversion equal to 3 over the out‐of‐sample period (October 1999 to July 2006). Note that ‘GR’ stands for the weight put in Russell 1000 Growth and ‘VL’ stands for the weight put in Russell 1000 Value. The horizontal axis shows the out‐of‐sample period, and the vertical axis shows optimal portfolio weights.

FIGURE 10.8 Optimal

unconstrained

normal and Gumbel portfolio weights for an investor with the degree of relative risk aversion equal to 7 over the out‐of‐sample period (October 1999 to July 2006). Note that ‘GR’ stands for the weight put in Russell 1000 Growth and ‘VL’ stands for the weight put in Russell 1000 Value. The horizontal axis shows the out‐of‐sample period, and the vertical axis shows optimal portfolio weights.

Chapter 11

FIGURE 11.1 Empirical relation between the US market and industry indices. This figure plots monthly excess returns for the US market vs. 12 constituent industry indices from July 1963 to December 2010 (in excess of the US 1‐month T‐bill). The boxed regions highlight threshold return values above +20% and below −20% for the industry indices and the US market.

FIGURE 11.2 Plots illustrating alternative dependence structures. This figure shows a variety of dependence structures for the bivariate case of

x

1

and

x

2

asset returns. (a) The circular dependence structure when the Pearson's correlation and Clayton copula parameters are close to zero. (b) The elliptical dependence structure produced for a correlation parameter of 0.5. (c) The asymmetric (lower‐tail) dependence structure produced for a Clayton copula parameter of 0.9.

FIGURE 11.3 Alternative efficient frontiers (annual

E

(

R

) vs. CVaR). This figure shows the annual expected returns against CVaR for portfolios of 3 and 12 assets. Nine alternative strategies are implemented for each portfolio using the full sample from July 1963 to December 2010. Clayton SC and Clayton CVC models allow for asymmetric dependence. Models that allow for elliptical dependence structures are the Gaussian, Student

t

copulas and MVN models. The Skew‐T model allows for asymmetries within the marginals while the Normal model does not.

FIGURE 11.4 Pattern of wealth accumulation for out‐of‐sample copula‐based portfolio strategies. This figure shows the accumulation of wealth from an initial hypothetical investment of $100 in each portfolio strategy at the start of the out‐of‐sample period for 3‐asset and 12‐asset portfolios. SC‐N is the Clayton standard copula (SC) with normal marginals, CVC‐N is the Clayton canonical vine copula (CVC) with normal marginals, SC‐S is the Clayton SC with Skew‐T marginals, CVC‐S is the Clayton CVC with Skew‐T marginals and MVN is the multivariate normal model (benchmark case).

FIGURE 11.5 Annual Sharpe ratio and cumulative return, out‐of‐sample, for CVC‐S vs. MVN models. This figure shows annual comparisons between the Sharpe ratio and annual cumulative returns between the CVC‐S and MVN models. CVC‐S is the Clayton canonical vine copula (CVC) with Skew‐T marginals and MVN is the multivariate normal model (benchmark case).

FIGURE 11.6 Difference in end‐of‐year portfolio value, out‐of‐sample, for CVC‐S vs. MVN models. This figure shows the difference in end‐of‐year portfolio values between CVC‐S and MVN annually. The end‐of‐year portfolio value of MVN is subtracted from CVC‐S, based on a hypothetical investment of $100 in each strategy at the beginning of each year. CVC‐S is the Clayton canonical vine copula (CVC) with Skew‐T marginals and MVN is the multivariate normal model (benchmark case).

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Asymmetric Dependence in Finance

Diversification, Correlation and Portfolio Management in Market Downturns

EDITED BY

JAMIE ALCOCK

STEPHEN SATCHELL

This edition first published 2018

© 2018 John Wiley & Sons Ltd

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Names: Alcock, Jamie, 1971– author. | Satchell, S. (Stephen) author.

Title: Asymmetric dependence in finance : diversification, correlation and portfolio management in market downturns / Jamie Alcock, Stephen Satchell.

Description: Hoboken : Wiley, 2018. | Series: Wiley finance | Includes bibliographical references and index. |

Identifiers: LCCN 2017039367 (print) | LCCN 2017058043 (ebook) | ISBN 9781119289029 (epub) | ISBN 9781119289012 (hardback) | ISBN 9781119289005 (ePDF) | ISBN 9781119288992 (e-bk)

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Cover Design: Wiley

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To the memory of John Knight

About the Editors

Dr Jamie Alcock is Associate Professor of Finance at the University of Sydney Business School. He has previously held appointments at the University of Cambridge, Downing College Cambridge and the University of Queensland. He was awarded his PhD by the University of Queensland in 2005. Dr Alcock's research interests include asset pricing, corporate finance and real estate finance. Dr Alcock has published over 40 refereed research articles and reports in high‐quality international journals. The quality of Dr Alcock's research has been recognized through multiple international research prizes, including most recently the EPRA Best Paper prize at the 2016 European Real Estate Society conference.

Stephen Satchell is a Life Fellow at Trinity College Cambridge and a Professor of Finance at the University of Sydney. He is the Emeritus Reader in Financial Econometrics at the University of Cambridge and an Honorary Member of the Institute of Actuaries. He specializes in finance and econometrics, on which subjects he has written at least 200 papers. He is an academic advisor and consultant to a wide range of financial institutions covering such areas as actuarial valuation, asset management, risk management and strategy design. Satchell's expertise embraces econometrics, finance, risk measurement and utility theory from both theoretical and empirical viewpoints. Much of his research is motivated by practical issues and his investment work includes style rotation, tactical asset allocation and the properties of trading rules, simulation of option prices and forecasting exchange rates.

Dr Satchell was an Academic Advisor to JP Morgan Asset Management, the Governor of the Bank of Greece and for a year in the Prime Minister's department in London.

Introduction

Asymmetric dependence (hereafter, AD) is usually thought of as a cross‐sectional phenomenon. Andrew Patton describes AD as ‘stock returns appear to be more highly correlated during market downturns than during market upturns’ (Patton, 2004).1 Thus, at a point in time when the market return is increasing, we might expect to find the correlation between any two stocks to be, on average, lower than the correlation between those same two stocks when the market return is negative. However, the term can also have a time‐series interpretation. Thus, it may be that the impact of the current US market on the future UK market may be quantitatively different from the impact of the current UK market on the future US market. This is also a notion of AD that occurs through time. Whilst most of this book addresses the former notion of AD, time‐series AD is explored in Chapters 4 and 7.

Readers may think that discussion of AD commenced during the Global Financial Crisis (GFC) of 2007–2009, however scholars have been exploring this topic in finance since the early 1990s. Mathematical statisticians have investigated asymmetric asymptotic tail dependence for much longer. The evidence thus far has found that the cross‐sectional correlation between stock returns has generally been much higher during downturns than during upturns. This phenomenon has been observed at the stock and the index level, both within countries and across countries. Whilst less analysis of time‐series AD with relation to market states has been carried out, it is highly likely that the results for time‐series AD will depend upon the frequency of data observation and the conditioning information set, inter alia.

The ideas behind the measurement of AD depend upon computing correlations over subsets of the range of possible values that returns can take. Assuming that the original data comes from a constant correlation distribution, once we truncate the range of values, the conditional correlation will change. This is the idea behind one of the key tools of analysis, the exceedance correlation. To understand the power of this technique, readers should consult Panels A and B on p. 454 of Ang and Chen (2002).2 The distributional assumptions for the data generating process now become critical. It can be shown that, as we move further into the tails, the exceedance correlation for a multivariate normal distribution tends to zero. Intuitively, this means that multivariate normally distributed random variables approach independence in the tails. Empirical plots in the analysis of AD tend to suggest that, in the lower tail at least, the near independence phenomenon does not occur. Thus we are led to consider other distributions than normality, an approach addressed throughout this book.

The most obvious impact of AD in financial returns is its effect on risk diversification. To understand this, we look at quantitative fund managers whose behaviour is described as follows. They typically use mean‐variance analysis to model the trade‐off between return and risk. The risk (variance) of a portfolio will depend upon the variances and correlations of the stocks in the portfolio. Optimal investments are chosen based on these numbers. One feature of such mean‐variance strategies is that one often ends up investing in a small number of funds and all other risks are diversified away as idiosyncratic correlations will average out. However, if these correlations tend to one then the averaging process will not eliminate idiosyncratic risks, diversification fails and the optimal positions chosen are no longer optimal. Said another way, risk will be underestimated and hedging strategies will no longer be effective.

The example above is just one case where AD will affect financial decision making. To the extent that AD influences the optimal portfolios of investors, it will clearly also affect the allocation of capital within the broader market and hence the cost of that capital to corporate entities. An understanding of AD as a financial phenomenon is not only important to financial risk managers but also to other senior executives in organizations. Solutions for managing AD are scarce, however Chapter 5 provides some answers to these problems.

This book looks at explanations for the ubiquitous nature of AD. One explanation that is attractive to economists is that AD derives from the preferences (utility functions) of individual market agents. Whilst quadratic preferences typically lead to relatively symmetric behaviour, theories such as loss aversion or disappointment aversion give expected utilities that have built‐in asymmetries with respect to future wealth. These preferences and their implications are discussed in Chapter 1. Such structures lead to the pricing of AD, and coupled with suitable dynamic processes for prices will generate AD that, theoretically at least, could be observed in financial markets. Chapter 3 explores the pricing of AD within the US equities market. These chapters discuss non‐linearity in utility as a potential source of AD. Another approach that will give similar outcomes is to model the dynamic price processes in non‐linear terms. Such an approach is carried out in Chapters 2 and 4.

It is understood that the origins of AD may well have a basis in individual and collective utility. This idea is investigated in Chapter 1, where Jamie Alcock and Anthony Hatherley explore the AD preferences of disappointment‐averse investors and how these preferences filter into asset pricing. One of the advantages of the utility approach is that it can be used to define gain and loss measures. The authors develop a new metric to capture AD based upon disappointment aversion and they show how it is able to capture AD in an economic and statistically meaningful manner. They also show that this measure is better able to capture AD than commonly used competing methods. The theory developed in this chapter is subsequently utilized in various ways in Chapters 3 and 9.

One explanation of AD is based on notions of non‐linear random variables. Stephen Satchell and Oliver Williams use this framework in Chapter 2 to build a model of a market where an option and a share are both traded, and investors combine these instruments into portfolios. This will lead to AD on future prices. The innovation in this chapter is to use mean‐variance preferences that add a certain amount of tractability. This model is then used to assess the factors that determine the size of the commodity trading advisor (CTA) market. This question is of some importance, as CTA returns seem to have declined as the volume of funds invested in them has increased. The above provides another explanation of the occurrence of AD.

In Chapter 3